Re: infinity

In article <MPG.1d877fa772bed6e798a1da@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Dik T. Winter said:
> > In article <MPG.1d7fc12f5eeb6e7898a1b1@xxxxxxxxxxxxxxxxxxxxxxxxx>
> > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> > > stephen@xxxxxxxxxx said:
> > ...
> > > > Of course you also deny that there is a maximum string length,
> > > > so I have no idea what S^L is supposed to mean.
> > > Given any string length and alphabet, that is the maximum number
> > > of unique srings in the language.
> >
> > That is obviously wrong. Let's have the alphabet consist of the
> > digits 1 and 2. Let us further define the value of a finite string
> > of length l as sum{i = 1..l} d_i * 2^(l - i). The language
> > consists of all strings of finite length. What is the L in this
> > case?

> For each L, there are 2^L strings of that length, and sum(x=0->L:
> 2^x) strings up to that length. If L is finite, then sum(x=0->L: 2^x)
> is a sum of a finite number of terms, each of which is finite, so
> there are a finite number of strings in a language with strings of
> finite length.

If TO meant 'bounded length' instead of 'finite length' he would be
right, but merely requiring 'finite length' allows the finite lengths to
be unboundedly large which allows the languages also to be unboundedly
large, larger than any finite quantity, and thus not finite!


> >
> > > Darryl just gave a similar "proof". When you say "for all finite
> > > k" then you are saying there is some upper bound to k.
> >
> > No. The first does *not* imply the second. Each and every k is
> > finite, but the sequence of k's is unbounded.

> Unbounded meaning potentially infinite.

If, by "potentially infinite" TO means inclusive of infinite values,
then he is wrong again. "Unbounded" of a set of numbers means that there
is no finite number as large as EVERY member of the set. It does not
mean anything other than that.

> For every finite maximum
> string length, there is a finite maximum language size. Only when
> string length is allowed to become infinite can the langauge be
> infinite, unless you have an infinite alphabet.

WRONG! AGAIN! If the alphabet size of the word lenght is unbounded,
though each consists of only finite values, so is the language size
unbounded, though it will also consist of only finite values.

Unbounded sets of numbers are Cantor-infinite.



> >
> > > What is the upper bound for string length?
> >
> > There is none.
> And yet, you reject the notion of infinite strings?

The members of a set of numbers can be unbounded without that set
containing any "infinite numbers". The set of "finite" naturals is the
prime example.

Is TO able to give any SPECIFIC finite upper bound of the set of "finite
naturals"? Until he can do so, finite string lengths can be unbounded,
with no need for any infinite strings.
.

Relevant Pages

  • TOs quantifier dyslexia rears its ugly head again
    ... >>> If the word length is finite, then the language is also finite. ... >> If the word length and alphabet size have finite upper bounds then ... > finite length and saying that string length has a finite upper bound. ... injects the set of finite naturals onto a proper subset of itself, ...
    (sci.math)
  • Re: infinity
    ... >>> If the word length is finite, then the language is also finite. ... >> If the word length and alphabet size have finite upper bounds then ... > finite length and saying that string length has a finite upper bound. ... injects the set of finite naturals onto a proper subset of itself, ...
    (sci.math)
  • Re: infinity
    ... >>> If the word length is finite, then the language is also finite. ... >>> x, being the length of the longest word in the language, and any finite ... > finite, that means none are infinite, therefore S^L is not infinite either. ... the maximum string length, i.e. the largest finite natural number. ...
    (sci.math)
  • Re: infinity
    ... >>> Why are you assuming that there is a longest word in the language? ... >> finite, that means none are infinite, therefore S^L is not infinite either. ... > I am not assuming that there is a longest word. ... > the maximum string length, i.e. the largest finite natural number. ...
    (sci.math)
  • Re: infinity
    ... >>> If the word length HAS AN UPPER BOUND, ... and I am not saying that a finite language is necessarily small. ... >> if the string length is strictly finite, ... Sounds infinite to me, this string elongation process, eh? ...
    (sci.math)